Flips and Terminations

9800807 Shokurov该奖项支持Shokurov和Kachi教授的一个项目。本项目主要研究高维对数最小模型程序中存在的问题，主要是4维及更高维对数翻转的存在问题。目的是从LMMPs本身和一些较低维度的终止中感应地获得这样的翻转。特别地，我们的目的是构造对数翻转，甚至是完成4维的线性矩阵规划。此外，还将给出族中翻转的几何描述。Shokurov和Kachi还将开始调查所需的终止，并展示LMMP的进一步应用。这是在代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的图形可以在平面上由最简单的方程定义，即多项式，或在3-空间中由最简单的几何结构给出，例如二次曲线。传统上，这个领域与数学的大多数分支相互作用。如今，这个领域不仅使用代数的方法，而且使用分析、拓扑学和数学物理的方法，相反，在这些领域以及数论、物理学、理论计算机科学和机器人学中都有应用。

9800807 Shokurov This award support a project of Professors Shokurov and Kachi. The project is focused on problems in the Log Minimal Model Program (LMMP) in higher dimensions, mainly, on the existence of the log flips in dimension 4 and higher. The aim is to obtain such flips inductively from the LMMP itself and some terminations in lower dimensions. In particular, the aim is to construct the log flips and even to complete the LMMP in dimension 4. In addition, a geometric description of flips in families will be given. Shokurov and Kachi will also start to investigate needed terminations and to demonstrate further applications of the LMMP. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials, or be given in the 3-space by the simplest geometric constructions, e. g., conic sections. Traditionally, the field interacts with most of branches of mathematics. Nowadays the field makes use of methods not only from algebra, but from analysis, topology and mathematical physics, and conversely is finding application in those fields as well as in number theory, physics, theoretical computer science, and robotics.